Finite Group Databasebeta

← #27141815

Subgroups — #27141815

Order 12416.

316 conjugacy classes of subgroups; only the 81 normal subgroups are shown.

normal subgroups only

This group has 316 conjugacy classes of subgroups — too many to draw. Only its 81 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C4⋊C4×D97C211yesmaximal, characteristic
2#20260698C212yesmaximal
3#20260504C211yesmaximal, characteristic, complemented
4#20260503C212yesmaximal, complemented
5#343×C97C211yesmaximal, characteristic, complemented
6C22×C4×D97C412yes—
7C22×C4×D97C2211yescharacteristic
8C2×#13095495C2212yes—
9C2×C4⋊Dic97C2211yescharacteristic
10C2×#13095490C412yes—
11C4⋊C4×D97—22no—
12C4⋊C4×D97—22no—
13#15690722—22no—
14#15690722—22no—
15C2×M16×C97C2212yes—
16C2×C4⋊C4×C97C2211yescharacteristic
17C23×D97C2×C411yescharacteristic
18C22×Dic97C2×C411yescharacteristic
19C22×Dic97C2×C412yes—
20C2×C4×D97—21no—
21C2×C4×D97—22no—
22C2×C4×D97—21no—
23C2×C4×D97—22no—
24C2×C4×D97—22no—
25C2×C4×D97—44no—
26C2×C4×D97D412yes—
27C2×C4×D97D412yes—
28#13095495—22no—
29#13095495—22no—
30C2×#11715895—22no—
31C4⋊Dic97—22no—
32#13095490—24no—
33C22×C4×C97C2×C412yes—
34C22×C4×C97C2311yescharacteristic
35M16×C97—22no—
36M16×C97—22no—
37C2×C8×C97—22no—
38C4⋊C4×C97—22no—
39C23×C97C22×C411yescharacteristic
40C22×D97—21no—
41C22×D97—21no—
42C22×D97—22no—
43C22×D97—22no—
44C22×D97C22⋊C411yescharacteristic
45C22×D97C22⋊C411yescharacteristic
46C2×C4×C97—22no—
47C2×C4×C97—42no—
48C2×C4×C97C2×D412yes—
49C2×C4×C97#4111yescharacteristic
50C2×C4×C97#4112yes—
51C2×C4×C97#4111yescharacteristic
52C2×Dic97—21no—
53C2×Dic97—21no—
54C2×Dic97—22no—
55C2×Dic97—42no—
56C2×Dic97C22⋊C412yes—
57C4×D97—22no—
58C4×D97—22no—
59C4×D97—22no—
60C4×D97—22no—
61C4×D97—44no—
62C4×D97—44no—
63C8×C97—42no—
64#11715895—42no—
65C22×C97—21no—
66C22×C97—21no—
67C22×C97C2×C22⋊C411yescharacteristic
68C22×C97#11711yescharacteristic
69C22×C97#11711yescharacteristic
70C2×D97—24no—
71C2×D97—24no—
72C2×D97—21no—
73C2×D97—21no—
74C2×D97—21no—
75C2×D97—21no—
76C2×D97#10012yes—
77C2×D97#10112yes—
78C4×C97—42no—
79C4×C97#12312yes—
80C4×C97#12312yes—
81Dic97—22no—
82Dic97—42no—
83C2×C97—21no—
84C2×C97—21no—
85C2×C97#38111yescharacteristic
86C2×C97C2×#10011yescharacteristic
87C2×C97C2×#10111yescharacteristic
88D97—22no—
89D97—22no—
90#1821—971nomaximal, Sylow
91C97#182111yescharacteristic, complemented, Sylow
92C22×C4⋊C4—971no—
93#407—972no—
94#343—972no—
95#343—971no—
96#343D9711yescharacteristic, complemented
97C23×C4—971no—
98C23×C4—972no—
99C2×M16—972no—
100C2×M16C2×D9712yes—
101C2×C4⋊C4—972no—
102C2×C4⋊C4—971no—
103C2×C4⋊C4—1942no—
104C2×C4⋊C4—1942no—
105C2×C4⋊C4C2×D9711yescharacteristic
106C22⋊C8—1942no—
107C22⋊C8—1942no—
108C24—971no—
109C22×C4—971no—
110C22×C4—972no—
111C22×C4—972no—
112C22×C4—972no—
113C22×C4—3884no—
114C22×C4—1942no—
115C22×C4—1942no—
116C22×C4—1941no—
117C22×C4—1942no—
118C22×C4—1941no—
119C22×C4C4×D9712yes—
120C22×C4C22×D9711yescharacteristic
121M16—22no—
122M16—22no—
123M16—1942no—
124M16—1942no—
125C2×C8—22no—
126C2×C8—1942no—
127C4⋊C4—22no—
128C4⋊C4—1944no—
129C4⋊C4—1942no—
130C23—971no—
131C23—971no—
132C23—1942no—
133C23—1942no—
134C23—1941no—
135C23—1941no—
136C23C2×C4×D9711yescharacteristic
137C2×C4—22no—
138C2×C4—42no—
139C2×C4—972no—
140C2×C4—3884no—
141C2×C4—3884no—
142C2×C4—3882no—
143C2×C4—1942no—
144C2×C4—1941no—
145C2×C4—1941no—
146C2×C4—1942no—
147C2×C4—1942no—
148C2×C4—1942no—
149C2×C4—1942no—
150C2×C4D4×D9712yes—
151C2×C4#1309551011yescharacteristic
152C2×C4#1309551112yes—
153C2×C4#1309551211yescharacteristic
154C8—42no—
155C8—3882no—
156C22—21no—
157C22—21no—
158C22—972no—
159C22—972no—
160C22—1944no—
161C22—1944no—
162C22—1941no—
163C22—1941no—
164C22—1941no—
165C22—1941no—
166C22C22⋊C4×D9711yescharacteristic
167C22#1569076711yescharacteristic
168C22#1569076811yescharacteristic
169C4—42no—
170C4—3882no—
171C4—1942no—
172C4#1569079212yes—
173C4#1569079512yes—
174C2—21no—
175C2—21no—
176C2—1942no—
177C2—1942no—
178C2#2026059711yescharacteristic
179C2#100×D9711yescharacteristic
180C2#101×D9711yescharacteristic
subgroupG/H
centerC22C22⋊C4×D97
commutatorC23×C97C22×C4
FrattiniC22×C4C22×D97
Fitting#343×C97C2
socleC22×C97C2×C22⋊C4
prime radicalC23×D97C2×C4
cosocleC22×C4×C97C23
hypercenter#343D97
hypocenterC97#1821
LCSG ▷ C23×C97 ▷ C2×C97 ▷ C97
UCSC1 ◁ C22 ◁ C22×C4 ◁ #343

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
3. #20260504C2semidirect
4. #20260503C2semidirect
5. #343×C97C2semidirect
91. C97#1821semidirect
96. #343D97semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
C2×C4⋊C4×D972
#202606982
#202605042
#202605032
#343×C972
#182197